Geometry and Combinatorics in Arizona

Geometry and Combinatorics
Workshop at Northern Arizona University
January 4-5, 2004, Flagstaff, AZ

Preliminary Program

Sunday, January 4, 2004.

09:30am-10:30am Sergey Yuzvinsky, University of Oregon.

TBA

11:00am-12:00am Sergey Fomin, University of Michigan.

Cluster algebras of finite type.
This talk will provide a quick introduction into combinatorics of cluster algebras. It is based on joint work with Andrei Zelevinsky.
Cluster algebras are a class of commutative rings that include (homogeneous) coordinate rings of many classical varieties associated with a complex semisimple Lie group, such as double Bruhat cells, Grassmannians, base affine spaces, etc. The classification of cluster algebras of finite type turns out to be yet another instance of the Cartan-Killing classification. The combinatorics underlying a cluster algebra of finite type is governed by the generalized associahedron, a particular convex polytope constructed from the corresponding root system.

02:00pm-03:00pm Eva-Maria Feichtner, ETZ Zürich.

Abelianizing real diffeomorphic actions of finite groups.
In this talk we provide abelianizations of diffeomorphic actions of finite groups on smooth real manifolds. Wonderful models for (local) subspace arrangements as defined by De Concini and Procesi and a careful analysis of linear actions on real vector spaces are at the core of our construction. In fact, we show that our abelianizations have stabilizers isomorphic to elementary abelian 2-groups.
A lot of inspiration and intuition for our abelianization construction is drawn from studying the permutation action of the symmetric group $S_n$ on real n-dimensional vector space in detail. We discuss stabilizer distinguishing stratifications for the De Concini-Procesi model of the braid arrangement, and provide an algebro-combinatorial setup for describing stabilizers.
This is joint work with Dmitry Kozlov (KTH Stockholm).

03:30pm-04:30pm Graham Denham, University of Western Ontario.

The homotopy Lie algebra of an arrangement.
I will describe some recent work that aims to generalize the well-known lower central series formulas for an arrangement group to arrangements that are not Koszul. The main object of interest here is the homotopy Lie algebra, which we approach by homological, combinatorial methods and determine explicitly in certain cases. We find a suitable notion of a resonance variety for nonabelian local systems. However, we note that a "natural" nonabelian generalization of a theorem of Eisenbud, Popescu, and Yuzvinsky fails for some arrangements. This is joint work with Alex Suciu)

05:00pm-06:00pm Dean Serenevy, Northeastern University.

TBA

Monday, January 5, 2004.

9:00am- 10:00am Hal Schenck, Texas A&M

Chen ranks and the Bernstein-Gelfand-Gelfand correspondence
The Bernstein-Gelfand-Gelfand correspondence is an isomorphism between the category of linear free complexes over the exterior algebra E and the category of graded free modules over the symmetric algebra S. I'll give a concrete introduction to the BGG correspondence, discussing several examples in detail and showing how to compute using Macaulay2.
Then we'll apply BGG to study the Chen ranks conjecture of Cohen-Suciu, which gives values for the Chen ranks of the fundamental group G of an arrangement complement in terms of the resonance variety R^1(A). Using BGG, we translate the conjecture into a question about the free resolution of the Orlik-Solomon algebra over E. Using BGG and our work on resolutions of graphic arrangements, we prove the Chen ranks conjecture for graphic arrangements.
Finally, I'll sketch the main result: the Chen ranks are given (asymptotically) by a polynomial of degree equal to the dimension of R^1(A), and if h_r denotes the number of components of R^1(A) of (projective) dimension r and k>> 0, then $\theta_k(G) \ge (k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}$.
This is joint work with A. Suciu.

10:30am-11:30am Dmitry Kozlov , KTH Stockholm

Topological obstructions to graph colorings
For any two graphs G and H Lovasz has defined a cell complex Hom(G,H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovasz concerning these complexes with G a cycle of odd length. More specifically, we show that if Hom(C_{2r+1},G) is k-connected, then \chi(G) >= k+4. Our actual statement is somewhat sharper, as we find obstructions already in the non-vanishing of powers of certain Stiefel-Whitney characteristic classes. This is joint work with Eric Babson.

02:30pm-03:30pm Klaus Altmann (FU Berlin)

TBA

04:00pm-05:00pm Daniel Matei (U Tokyo)

Massey products of arrangement groups
Abstract [pdf].

05:30pm-06:30pm Richard Randell (U Iowa)

Finite-type invariants for singular line arrangements.
We observe that to any oriented line arrangement in real projective three-space one may associate finite-type invariants in one of the usual ways: first consider a Kauffman bracket skein relation and module, then pass to Jones polynomials and finite-type invariants for non-intersecting lines in the usual way via change of variables, then use orientation to split intersections. Thus we follow the path cleared for us by Kauffman and Bar-Natan on one hand, and by Przytycki, Viro and Drobotukhina on the other. We present basic properties and calculations of these invariants for a number of examples. We also consider the barriers to definition of Khovanov homology in this setting (considerable), relationships with configuration spaces (intimate), and connections with knot theory (inspirational).

For further information, please contact:

Michael Falk

Department of Mathematics
Northern Arizona University
Flagstaff, AZ, U.S.A.
e-mail: michael.falk@nau.edu

Eva Maria Feichtner

Department of Mathematics
ETH Zürich
8092 Zürich, Switzerland
e-mail: feichtne@math.ethz.ch

Dmitry Kozlov

Department of Mathematics
Royal Institute of Technology
10044 Stockholm, Sweden
e-mail: kozlov@math.kth.se

Back to the workshop main page.

09:30am-10:30am		Sergey Yuzvinsky, University of Oregon.
		TBA

11:00am-12:00am		Sergey Fomin, University of Michigan.
		Cluster algebras of finite type. This talk will provide a quick introduction into combinatorics of cluster algebras. It is based on joint work with Andrei Zelevinsky. Cluster algebras are a class of commutative rings that include (homogeneous) coordinate rings of many classical varieties associated with a complex semisimple Lie group, such as double Bruhat cells, Grassmannians, base affine spaces, etc. The classification of cluster algebras of finite type turns out to be yet another instance of the Cartan-Killing classification. The combinatorics underlying a cluster algebra of finite type is governed by the generalized associahedron, a particular convex polytope constructed from the corresponding root system.

02:00pm-03:00pm		Eva-Maria Feichtner, ETZ Zürich.
		Abelianizing real diffeomorphic actions of finite groups. In this talk we provide abelianizations of diffeomorphic actions of finite groups on smooth real manifolds. Wonderful models for (local) subspace arrangements as defined by De Concini and Procesi and a careful analysis of linear actions on real vector spaces are at the core of our construction. In fact, we show that our abelianizations have stabilizers isomorphic to elementary abelian 2-groups. A lot of inspiration and intuition for our abelianization construction is drawn from studying the permutation action of the symmetric group $S_n$ on real n-dimensional vector space in detail. We discuss stabilizer distinguishing stratifications for the De Concini-Procesi model of the braid arrangement, and provide an algebro-combinatorial setup for describing stabilizers. This is joint work with Dmitry Kozlov (KTH Stockholm).

03:30pm-04:30pm		Graham Denham, University of Western Ontario.
		The homotopy Lie algebra of an arrangement. I will describe some recent work that aims to generalize the well-known lower central series formulas for an arrangement group to arrangements that are not Koszul. The main object of interest here is the homotopy Lie algebra, which we approach by homological, combinatorial methods and determine explicitly in certain cases. We find a suitable notion of a resonance variety for nonabelian local systems. However, we note that a "natural" nonabelian generalization of a theorem of Eisenbud, Popescu, and Yuzvinsky fails for some arrangements. This is joint work with Alex Suciu)

05:00pm-06:00pm		Dean Serenevy, Northeastern University.
		TBA

9:00am- 10:00am		Hal Schenck, Texas A&M
		Chen ranks and the Bernstein-Gelfand-Gelfand correspondence The Bernstein-Gelfand-Gelfand correspondence is an isomorphism between the category of linear free complexes over the exterior algebra E and the category of graded free modules over the symmetric algebra S. I'll give a concrete introduction to the BGG correspondence, discussing several examples in detail and showing how to compute using Macaulay2. Then we'll apply BGG to study the Chen ranks conjecture of Cohen-Suciu, which gives values for the Chen ranks of the fundamental group G of an arrangement complement in terms of the resonance variety R^1(A). Using BGG, we translate the conjecture into a question about the free resolution of the Orlik-Solomon algebra over E. Using BGG and our work on resolutions of graphic arrangements, we prove the Chen ranks conjecture for graphic arrangements. Finally, I'll sketch the main result: the Chen ranks are given (asymptotically) by a polynomial of degree equal to the dimension of R^1(A), and if h_r denotes the number of components of R^1(A) of (projective) dimension r and k>> 0, then $\theta_k(G) \ge (k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}$. This is joint work with A. Suciu.

10:30am-11:30am		Dmitry Kozlov , KTH Stockholm
		Topological obstructions to graph colorings For any two graphs G and H Lovasz has defined a cell complex Hom(G,H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovasz concerning these complexes with G a cycle of odd length. More specifically, we show that if Hom(C_{2r+1},G) is k-connected, then \chi(G) >= k+4. Our actual statement is somewhat sharper, as we find obstructions already in the non-vanishing of powers of certain Stiefel-Whitney characteristic classes. This is joint work with Eric Babson.

02:30pm-03:30pm		Klaus Altmann (FU Berlin)
		TBA

04:00pm-05:00pm		Daniel Matei (U Tokyo)
		Massey products of arrangement groups Abstract [pdf].

05:30pm-06:30pm		Richard Randell (U Iowa)
		Finite-type invariants for singular line arrangements. We observe that to any oriented line arrangement in real projective three-space one may associate finite-type invariants in one of the usual ways: first consider a Kauffman bracket skein relation and module, then pass to Jones polynomials and finite-type invariants for non-intersecting lines in the usual way via change of variables, then use orientation to split intersections. Thus we follow the path cleared for us by Kauffman and Bar-Natan on one hand, and by Przytycki, Viro and Drobotukhina on the other. We present basic properties and calculations of these invariants for a number of examples. We also consider the barriers to definition of Khovanov homology in this setting (considerable), relationships with configuration spaces (intimate), and connections with knot theory (inspirational).

Geometry and Combinatorics Workshop at Northern Arizona University January 4-5, 2004, Flagstaff, AZ

Preliminary Program

Sunday, January 4, 2004.

Monday, January 5, 2004.

Geometry and Combinatorics
Workshop at Northern Arizona University
January 4-5, 2004, Flagstaff, AZ